Question

Find the following :-
${cos}^{ - 1} (cos( \frac{7\pi}{4}))$
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Answer 1

To find :

Value of $\sf {cos}^{-1} \left[cos \left( \frac{7\pi}{4}\right)\right]$

Solution :

$\implies\sf {cos}^{-1} \left[cos \left( \frac{7\pi}{4}\right)\right]$

As we know that range (principal value branch) of cos^-1 is [0, π], and 7π/4 is not coming in range. So we have to use some formula.

• Formula = cos(2π - ∅) = cos ∅

$\implies\sf {cos}^{-1}\left[cos \left( 2\pi-\dfrac{\pi}{4}\right)\right]$

• Now it came in range
• cos^-1(cos x) = x

$\implies\sf {cos}^{-1}\left[cos \left(\dfrac{\pi}{4}\right)\right]$

$\implies\sf \dfrac{\pi}{4}$

•°• The value of $\sf {cos}^{-1} \left[cos \left( \frac{7\pi}{4}\right)\right]$ is π/4

Answer 2

Step-by-step explanation:

$\tt \: new \cos( \frac{7\pi}{4} ) = \cos( \frac{\pi}{4} ) \\ \tt \: since \: {cos}^{ - 1} x \: is \: the \: inverse \: of \: cos \\ \tt \: then \: {cos}^{ - 1} (cos( \frac{\pi}{4} )) = \frac{\pi}{4}$